The reason for this problem is so transparent it's laughable: easy access to borrowed money naturally creates inflation in the target market until heavy borrowing becomes the norm. It happens again and again throughout history.
The only way to be truly on the risk-free side of this one though is to own a school.
As a principle, I would downsize or mortgage my house before I'd send my children into the world as debt slaves. For their part, there are plenty of state-funded schools where tuition is 10K/yr or less.
The whole thing is backwards: these lenders and schools are businesses and should be marketing and selling you their product, not forcing you to "apply."
If foreigners have the means to pay non-resident costs to attend our public schools, then I see no problem at all selling them an education. Giving them grants in lieu of citizens is just bad practice. Lending them money is bad risk management.